What I Learned Today: Scale Drawings and Maps

I asked my 15-year-old what she learned today at school. She paused for a moment and then answered my question by asking me what I learned at school today.

It took me a while to think about what I had learned [which will make me more patient when I ask her the question again tomorrow], and then I remembered and shared with her:

We are working with some teachers who are using the Illustrative Mathematics 6–8 Math curriculum. The 7^th grade teachers are in Unit 1, Scale Drawings. They are working with Scale Drawings and Maps. Today I learned to look more closely at the scale given for a map.

Look at the following for a moment. What’s the same? What’s different?

The last two are from Illustrative Mathematics, which you can download for free at openupresources.org.

What’s different about the scales on the last two?

Attend to precision, MP6, says, “Mathematically proficient students try to communicate precisely to others. … They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately.”

I’m not sure that we would have noticed a difference, except that we were trying to find some assessment items from another source and saw that many aligned to 7.G.A.1 included a scale in the form of “1 cm = 100 miles”. I’ve looked at lots of maps and I never noticed the incongruity of saying that 1 cm equals 100 miles. We don’t really mean that 1 cm equals 100 miles, right? Not in the same sense that we say 4 quarters equals $1 or 3+4=7. Is there any wonder that our students misuse the equal sign?

And so the journey continues, grateful for the authors of this curriculum who make me pay closer attention to attending to precision and grateful for my daughter who makes me think and share about what I’m learning, too …

 

MP6 – Defining Terms

How do you provide your students the opportunity to attend to precision?

Writing sound definitions is a good practice for students, making all of us pay close attention to what something is and is not.

I’ve learned from Jessica Murk about Bongard Problems, which give students practice creating sound definitions based on what something is and is not.

What can you say about every figure on the left of the page that is not true about every figure on the right side of the page? (Bongard Problem #16)

Last year when I asked students to define circle, I found it hard to select and sequence the responses that would best contribute to a whole class discussion without taking too much class time.

I remember reading Dylan Wiliam’s suggestion in Embedding Formative Assessment (chapter 6, page 147) to have students give feedback to student responses that aren’t from their own class. I think it’s still helpful for students to spend time writing their own definition, and possibly trying to break a partner’s definition, but I wonder whether using some of last year’s responses to drive a whole class discussion this year might be helpful.

• a shape with no corners
• A circle is a shape that is equal distance from the center.
• a round shape whose angles add up to 360 degrees
• A circle is a two-dimensional shape, that has an infinite amount of lines of symmetry, and a total of 360 degrees.
• A 2-d figure where all the points from the center to the circumference are equidistant.

We recently discussed trapezoids.

Based on the diagram, how would you define trapezoid?

Does how you define trapezoid depend on how you construct it?

Can you construct a dynamic quadrilateral with exactly one pair of parallel sides?

And so the #AskDontTell journey continues …

 

MP6 – Mapping a Parallelogram Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

We continued working on our learning intention: I can map a figure onto itself using transformations.

Perform and describe a [sequence of] transformation[s] that will map parallelogram ABCD onto itself.

This task also requires students to practice I can look for and make use of structure. What auxiliary objects will be helpful in mapping the parallelogram onto itself?

The student who shared her work drew the diagonals of the parallelogram so that she could use the intersection of the diagonals as the center of rotation.

Then she rotated the parallelogram 180˚ about that point.

Could you use only reflections to carry a parallelogram onto itself?

You can. How can you describe the sequence of reflections to carry the parallelogram onto itself?

How else could you carry a parallelogram onto itself?

 

MP6 – Mapping a Figure Onto Itself

How do you provide your students the opportunity to practice I can attend to precision?

Jill and I have worked on a leveled learning progression for MP6:

Level 4:

I can distinguish between necessary and sufficient language for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

CCSS G-CO 3:

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

Our learning intention for the day was I can map a figure onto itself using transformations.

Performing a [sequence of] transformation[s] that will map rectangle ABCD onto itself is not the same thing as describing a [sequence of] transformation[s].

We practiced both, but we focused on describing.

I asked the student who listed several steps to share his work.

1. rotate rectangle 180˚ about point A
2. translate rectangle A’B’C’D’ right so that points A’ and B line up as points B’ and A. [What vector are you using?]
3. Reflect rectangle A”B”C”D” onto rectangle ABCD to get it to reflect onto itself. [About what line are you reflecting?]

What if we want to carry rectangle ABCD onto rectangle CDAB? How is this task different from just carrying rectangle ABCD onto itself?

What about mapping a regular pentagon onto itself?

Many students suggested using a single rotation, but they didn’t note the center of rotation. How could you find the center of rotation for a single rotation to map the pentagon onto itself?

This student used the intersection of the perpendicular bisectors to find the center of rotation, but didn’t know what angle to use for the rotation. How would you find an angle of rotation that would work?

What can you do other than a single rotation?

This student reflected the pentagon about the perpendicular bisectors of one of the side of the pentagon.

The descriptions students gave made it obvious that we needed more work on describing. The next day, we took some of the descriptions and critiqued them. Which students have attended to precision?

It’s good work to distinguish precision from knowing what someone means as we learn to attend to precision. And so the journey continues …

 

What’s My Rule?

We practice “I can look for and make use of structure” and “I can look for and express regularity in repeated reasoning” almost every day in geometry.

This What’s My Rule? relationship provided that opportunity, along with “I can attend to precision”.

What rule can you write or describe or draw that maps Z onto W?

As students first started looking, I heard some of the following:

• positive x axis
• x is positive, y equals 0
• they come together on (2,0)
• (?,y*0)
• when z is on top of w, z is on the positive side on the x axis

 

Students have been accustomed to drawing auxiliary objects to make use of the structure of the given objects.

As students continued looking, I saw some of the following:

Some students constructed circles with W as center, containing Z. And with Z as center, containing W.

Others constructed circles with W as center, containing the origin. And with Z as center, containing the origin.

Others constructed a circle with the midpoint of segment ZW as the center.

Another student recognized that the distance from the origin to Z was the same as the x-coordinate of W.

And then made sense of that by measuring the distance from W to the origin as well.

Does the redefining Z to be stuck on the grid help make sense of the relationship between W and Z?

 

As students looked for longer, I heard some of the following:

• The length of the line segment from the origin to Z is the x coordinate of W.
• w=((distance of z from origin),0)
• The Pythagorean Theorem

Eventually, I saw a circle with the origin as center that contained Z and W.

I saw lots of good conversation starters for our whole class discussion when I collected the student responses.

And so, as the journey continues,

Where would you start?

What questions would you ask?

How would you close the discussion?

 

0.9 Repeating

I got to teach one of my favorite lessons in a Precalculus class this week, which I developed several years ago from a paper by Thomas Osler, Fun with 0.999…

We started with a Quick Poll. Students could select as many or as few choices as they wanted.

I shared their responses separated

and grouped together.

In the first class, one student selected all three choices.

In the second class, 5 students selected all three choices.

I set the timer for a few minutes and asked students to think individually about how they could argue their selection(s).

Then I asked them to talk together about their ideas.

I walked around and listened. These are the conversations I heard:

A: 1/3 is 0.3 repeating. 2/3 is 0.6 repeating. If we add 1/3 and 2/3, we get 1. If we add 0.3 repeating and 0.6 repeating, we get 0.9 repeating.

B: 1/9 is 0.1 repeating. If we multiply 1/9 by 1, we get 1. If we multiply 0.1 repeating by 9, we get 0.9 repeating.

C: 1/3 is 0.3 repeating. If we add 1/3 three times, we get 1. If we add 0.3 repeating three times, we get 0.9 repeating.

D: If x=0.9 repeating, then 10x=9.9 repeating. (It was clear that a few students had seen Vi Hart talk about 0.9 repeating. Even so, this was all they had for now.)

E: I think this is like Zeno’s Paradox. To walk across the room, you have to walk halfway, and halfway again, and halfway again.

This was the perfect opportunity to deliberately sequence the students’ thinking and let them make connections between their arguments (5 Practices style). With which conversation would you start?

We started with argument C. More than one person shook their head in disbelief, even though they agreed that the argument was convincing.

Next we moved to argument A, which was very similar to argument C.

Next we moved to argument B.

I had a few suggestions of what to do, based on the article from the AMATYC Review. We went to one of those next that the students hadn’t thought of: If x=0.9 repeating, what happens when you divide the equation by 3?

A student shared their work differently in each class, showing that x=1.

We moved next to argument D. Again, students shared their thinking differently in each class.

No one thought about Zeno’s Paradox in the first class. So I asked them how we could express 0.9 repeating as a sum.

And then I sent a Quick Poll to collect their responses.

In the second class, I asked the students with argument E to share their thoughts. They got at the infinite sum idea, so without decomposing 0.9 repeating as a class, I sent the Quick Poll. Lots of students came up with a sum that equaled 1. Only one of those was clearly 0.9+0.99 +0.999+…

(I didn’t show them the responses equal to 1 in green when I showed them their results.)

So we practiced look for and make use of structure together. How can we decompose 0.9 repeating into a sum?

I sent the poll again.

We concluded the lesson by polling the first question again. In the first class, 4 additional students believed only that 0.9 repeating = 1 at the end.

In the second class the number of students selecting only choice A changed from 6 to 13.

Our #AskDontTell journey continues, one lesson at a time …

 

The Equation of a Circle

Expressing Geometric Properties with Equations

G-GPE.A Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

How do you provide an opportunity for your students to make sense of the equation of a circle in the coordinate plane? We recently use the Geometry Nspired activity Exploring the Equation of a Circle.

Students practiced look for and express regularity in repeated reasoning. What stays the same? What changes?

It’s a right triangle.

The hypotenuse is always 5.

The legs change.

What else do you notice? What has to be true for these objects?

The Pythagorean Theorem works.

How?

Leg squared plus leg squared equals five squared.

What do you notice about the legs? How can we represent the legs on the graph?

One leg is always horizontal.

One leg is always vertical.

How can we represent their lengths in the coordinate plane?

x and y?

(I think they thought that the obvious was too easy.)

What do x and y have to do with point P?

Oh! They’re the x- and y-coordinates of point P.

So what can we say is always true?

Is there an equation that is always true?

x²+y²=5²

What path does P travel? (This was preceded by – I’m going to ask a question, but I don’t want you to answer out loud. Let’s give everyone time to think.)

And then we traced point P as we moved it about coordinate plane.

So P makes a circle, and we have figured out that the equation of that circle is x²+y²=5².

I then let them explore two other pages with their teams, one where they could change the radius of the circle and one where they could change the center of the circle.

And then they answered a few questions about what they found. I used Class Capture to watch as they practiced look for and express regularity in repeated reasoning.

Here are the results of the questions that they worked.

What would you do next?

What I didn’t do at this point was differentiate my instruction. It occurred to me as soon as I got the results that I should have had a plan of what to do with the students who got 1 or 2 questions correct. It turns out that it was a team of students – already sitting together – who needed extra support – but I didn’t figure that out until later. Luckily, my students know that formative assessment isn’t just for me, the teacher – it’s for them, too. They share the responsibility in making a learning adjustment before the next class when they aren’t getting it.

We pressed on together – to make more sense out of the equation of a circle. I used a few questions from the Mathematics Assessment Project formative assessment lesson, Equations of Circles 1, getting at specific points on the circle.

And then I wondered whether we could begin making a circle. I assigned a different section of the x-y coordinate plane to each team. Send me a point (different from your team member) that lies on the circle x²+y²=64. Quadrant II is a little lacking, but overall, not too bad.

How can we graph the circle, limited to functions?

How can we tell which points are correct?

I asked them to write the equation of a circle given its center and radius, practicing attend to precision.

54% of the students were successful. The review workspace helps us attend to precision as well, since we can see how others answered.

(At the beginning of the next class, 79% of the students could write the equation, practicing attend to precision.)

I have evidence from the lesson that students are building procedural fluency from conceptual understanding (one of the NCTM Principles to Actions Mathematics Teaching Practices).

But what I liked best is that by the end of the lesson, most students reached level 4 of look for and express regularity in repeated reasoning: I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

When I asked them the equation of a circle with center (h,k) and radius r, 79% told me the standard form (or general for or center-radius form, depending on which textbook/site you use) instead of me telling them.

We closed the lesson by looking back at what happens when the circle is translated so that its center is no longer the origin. How does the right triangle change? How can that help us make sense of equation of the circle?

And so the journey continues, one #AskDontTell learning episode at a time.

 

SMP6: Attend to Precision #LL2LU

We want every learner in our care to be able to say

I can attend to precision.

CCSS.MATH.PRACTICE.MP6

But what if I can’t attend to precision yet? What if I need help? How might we make a pathway for success?

 

Level 4:
I can distinguish between necessary and sufficient conditions for definitions, conjectures, and conclusions.

Level 3:
I can attend to precision.

Level 2:
I can communicate my reasoning using proper mathematical vocabulary and symbols, and I can express my solution with units.

Level 1:
I can write in complete mathematical sentences using equality and inequality signs appropriately and consistently.

 

How many times have you seen a misused equals sign? Or mathematical statements that are fragments?

A student was writing the equation of a tangent line to linearize a curve at the point (2,-4).

He had written

y+4=3(x-2)

And then he wrote:

He absolutely knows what he means: y=-4+3(x-2).

But that’s not what he wrote.

 

Which student responses show attention to precision for the domain and range of y=(x-3)2+4? Are there others that you and your students would accept?

How often do our students notice that we model attend to precision? How often to our students notice when we don’t model attend to precision?

Attend to precision isn’t just about numerical precision. Attend to precision is also about the language that we use to communicate mathematically: the distance between a point and a line isn’t just “straight” – it’s the length of the segment that is perpendicular from the point to the line. (How many times have you told your Euclidean geometry students “all lines are straight”?)

But it’s also about learning to communicate mathematically together – and not just expecting students to read and record the correct vocabulary from a textbook.

[Cross posted on Experiments in Learning by Doing]

 

Classifying Triangles

We look specifically at 45-45-90 triangles on the first day of our Right Triangles unit. I’ve already written specifically about what the 45-45-90 exploration looked like, but I wanted to note a conversation that we had before that exploration.

Jill and I had recently talked about introducing new learning by drawing on what students already know. I’ve always started 45-45-90 triangles by having students think about what they already know about these triangles (even though many have never called them 45-45-90 triangles before). After hearing about one of Jill’s classes, though, I started by asking students to make a column for triangles, right triangles, and equilateral triangles, noting what they know to always be true for each. This short exercise gave students the opportunity to attend to precision with their vocabulary.

  

It occurred to me while we were talking that having students draw a Venn Diagram to organize triangles, right triangles, and equilateral triangles might be an interesting exercise. How would you draw a Venn Diagram to show the relationship between triangles, right triangles, and equilateral triangles?

In my seconds of anticipating student responses, I expected one visual but got something very different.

  

What does it mean for an object to be in the intersection of two sets? Or the intersection of three sets? Or in the part of the set that doesn’t intersect with the other sets?

   

Then we thought specifically about 45-45-90 triangles. What do you already know? Students practiced look for and make use of structure.

One student suggested that the legs are half the length of the hypotenuse. Instead of saying that wouldn’t work or not writing it on our list, I added it to the list and then later asked what would be the hypotenuse for a triangle with legs that are 5.

10.

I wrote 10 on the hypotenuse and waited.

But that’s not a triangle?

What?

5-5-10 doesn’t make a triangle.

Why not?

It would collapse (students have a visual image for a triangle collapsing from our previous work on the Triangle Inequality Theorem).

Does the Pythagorean Theorem work for 5-5-10?

Students reflected the triangles about the legs and hypotenuse to compose the 45-45-90 triangle into squares and rectangles.

 

And they constructed an altitude to the hypotenuse to decompose the 45-45-90 triangle into more 45-45-90 triangles.

And then we focused on the relationship between the legs and the hypotenuse using the Math Nspired activity Special Right Triangles.

And so the journey continues … listening to and learning alongside my students.

 

Proving Segments Congruent First

CCSS-M.G-CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Proving triangle congruence from rigid motions has been one of our most challenging new standards. Which is exciting for me as a teacher, because I’m always up for learning about something that all of the textbooks from which I’ve taught geometry have let slide into our deductive system as postulates with no need of proof.

So after two years of teaching these standards, it occurred to me that maybe we shouldn’t start with proving triangle congruence using rigid motions. Instead, why don’t we see what happens when we start with two segments.

What set of rigid motions will show that segment AB is congruent to segment CD?

Students started creating a plan (sequence of rigid motions) on paper. But before we moved to the technology to test the plans, we talked about attend to precision. Instead of saying that you’ll use a translation and a rotation, let’s be specific about what translation and what rotation. A translation of what segment by what vector? A rotation of what segment about what point using what angle measure?

I began to see the specifics on paper, but while students were pretty confident about the translations they had named, they were not totally confident about the rotations they had named. We needed our technology to help us see.

One team translated segment AB using vector BC.

Then they rotated segment A’B’ about B’ using angle A’B’D.

We can see that it works: the blue pre-image is now black. And we can move the original segment to see that it sticks.

Did this team prove that segment AB is congruent to segment CD? Or did they prove that segment AB is congruent to segment DC? Does it matter?

How many times have we told students that saying segment AB is congruent to segment CD is the same thing as saying that segment AB is congruent to segment DC? It occurred to me in the midst of this lesson that we have actually shown why those endpoints are interchangeable in our congruence statement for segments.

Another team used a translation (segment AB using vector BD) and was trying to use a reflection. When I discussed their work with them, they said, “we know where to draw the line, but we don’t know how to describe it”.

That team presented their work to the class using Live Presenter.

They drew in a line and reflected segment A’B’ about the line.

 

It didn’t work, but they moved the line into the right place.

So what’s significant about the line of reflection that works?

Someone in the class suggested that it’s the angle bisector of angle DCB”.

Is it?

We don’t have to wonder. We can verify using our Angle Bisector tool.

What else is significant about the line?

This year in geometry we are often going to have to see what isn’t pictured (look for and make use of structure).

What if we draw in the auxiliary segment B’D? What else is significant about the line of reflection?

It’s the perpendicular bisector of segment B’D. And again, we don’t have to wonder. We can verify using our Perpendicular Bisector tool.

Can you show that segment AB is congruent to segment CD using only reflections?

We left this exercise for Problem Solving Points, as of course by now we were running out of class time.

One student shared her work with me the next day.

And the next.

We still have work to do. But that’s good … this was the first week of school.

We did this task at my CMC-S session recently, and I asked about using at least one reflection since, like my students, most everyone translated and then rotated.

I was expecting to hear: Reflect segment AB about the perpendicular bisector of segment AC.

Then reflect segment A’B’ about the perpendicular bisector of segment A’D.

Some day I’m going to learn to not be surprised by solutions that are different from mine. One of the participants suggested extending lines AB and CD until they meet at point I. Then reflecting segment AB about the angle bisector of angle BIC. Then they translated segment A’B’ using vector B’C. Several weren’t satisfied with proving segment AB congruent to segment DC, so we noted that we could reflect segment B’’A’’ about its perpendicular bisector to show that segment AB is congruent to segment CD.

 

 

Another participant asked whether students had been confused moving into proving triangles congruent by changing the order of the vertices, since we can’t do that in a congruence statement. We did this at the beginning of the rigid motions unit, and I didn’t notice any issues moving into congruence statements for triangles where the order of the vertices matters.

And so the journey continues … every once in a while figuring out a task that will help us along the way towards meeting our learning goals.